Robustness and diffusion of pointer states

Classical properties of an open quantum system emerge through its interaction with other degrees of freedom (decoherence). We treat the case where this interaction produces a Markovian master equation for the system. We derive the corresponding distinguished local basis (pointer basis) by three methods. The first demands that the pointer states mimic as closely as possible the local nonunitary evolution. The second demands that the local entropy production be minimal. The third imposes robustness on the inherent quantum and emerging classical uncertainties. All three methods lead to localized Gaussian pointer states, their formation and diffusion being governed by well-defined quantum Langevin equations.     

Coupling-induced bipartite pointer states in arrays of electron billiards: quantum Darwinism in action?

We discuss a quantum system coupled to the environment, composed of an open array of billiards (dots) in series. Beside pointer states occurring in individual dots, we observe sets of robust states which arise only in the array. We define these new states as bipartite pointer states, since they cannot be described in terms of simple linear combinations of robust single-dot states. The classical existence of bipartite pointer states is confirmed by comparing the quantum-mechanical and classical results. The ability of the robust states to create "offspring" indicates that quantum Darwinism is in action.     

Open quantum dots: II. Probing the classical to quantum transition

Open quantum dots provide a natural system in which to study both classical and quantum features of transport. From the classical point of view these dots possess a mixed phase space which yields families of closed, regular orbits as well as an expansive sea of chaos. An important question concerns the manner in which these classical states evolve into the set of quantum states that populate the dot in the quantum limit. In the reverse direction, the manner in which the quantum states evolve to the classical world is governed strongly by Zurek's decoherence theory. This was discussed from the quantum perspective in an earlier review?(Ferry et?al 2011 Semicond. Sci. Technol. 26 043001). Here, we discuss the nature of the various classical states, how they are formed, how they progress to the quantum world, and the signatures that they create in magnetotransport and general conductance studies of these dots.     

Preferred states of open electronic systems

System-environment interaction may introduce dynamic destruction of quantum coherence, resulting in a special representation named as pointer states. In this work, pointer states of an open electronic system are studied. The decoherence effect is taken into account through two different ways which are Büttiker's virtual probe model and strong electron-phonon interaction in the polaron picture. The pointer states of the system with different coupling strength are investigated. The pointer states are identified by tracking the eigenstates of the density matrix in real-time propagation. It is found that the pointer states can emerge for arbitrary coupling strength. And the pointer states deform to the eigenstates of the system in the strong coupling limit, which indicates the vanish of quantumness in the strong coupling limit.     

Localization of Bose-Einstein condensates in optical lattices

The dynamics of repulsive bosons condensed in an optical lattice is effectively described by the Bose-Hubbard model. The classical limit of this model, reproduces the dynamics of Bose-Einstein condensates, in a periodic potential, and in the superfluid regime. Such dynamics is governed by a discrete nonlinear Schrödinger equation. Several papers, addressing the study of the discrete nonlinear Schrödinger dynamics, have predicted the spontaneous generation of (classical) breathers in coupled condensates. In the present contribute, we shall focus on localized solutions (quantum breathers) of the full Bose-Hubbard model. We will show that solutions exponentially localized in space and periodic in time exist also in absence of randomness. Thus, this kind of states, reproduce a novel quantum localization phenomenon due to the interplay between bounded energy spectrum and non-linearity.     

Classical Mechanics Is not the ħ, → 0 Limit of Quantum Mechanics

Both the set of quantum states and the set of classical states described by symplectic tomographic probability distributions (tomograms) are studied. It is shown that the sets have a common part but there exist tomograms of classical states which are not admissible in quantum mechanics and, vice versa, there exist tomograms of quantum states which are not admissible in classical mechanics. The role of different transformations of reference frames in the phase space of classical and quantum systems (scaling and rotation) determining the admissibility of tomograms as well as the role of quantum uncertainty relations are elucidated. The union of all admissible tomograms of both quantum and classical states is discussed in the context of interaction of quantum and classical systems. Negative probabilities in classical and quantum mechanics corresponding to tomograms of classical and quantum states are compared with properties of nonpositive and nonnegative density operators, respectively. The role of the semigroup of scaling transforms of the Planck's constant is discussed.     

Emergence of classicality for primordial fluctuations: Concepts and analogies

We clarify the way in which cosmological perturbations of quantum origin, produced during inflation, assume classical properties. Two features play an important role in this process: First, the dynamics of fluctuations which are presently on large cosmological scales leads to a very peculiar state (highly squeezed) that is indistinguishable, in a precise sense, from a classical stochastic process. This holds for almost all initial quantum states. Second, the process of decoherence by interaction with the environment distinguishes the field amplitude basis as the robust pointer basis. We discuss in detail the interplay between these features and use simple analogies such as the free quantum particle to illustrate the main conceptual issues.     

A Dynamics for Discrete Quantum Gravity

This paper is based on the causal set approach to discrete quantum gravity. We first describe a classical sequential growth process (CSGP) in which the universe grows one element at a time in discrete steps. At each step the process has the form of a causal set (causet) and the “completed” universe is given by a path through a discretely growing chain of causets. We then quantize the CSGP by forming a Hilbert space H on the set of paths. The quantum dynamics is governed by a sequence of positive operators ρ _n on H that satisfy normalization and consistency conditions. The pair (H,{ρ _n }) is called a quantum sequential growth process (QSGP). We next discuss a concrete realization of a QSGP in terms of a natural quantum action. This gives an amplitude process related to the “sum over histories” approach to quantum mechanics. Finally, we briefly discuss a discrete form of Einstein’s field equation and speculate how this may be employed to compare the present framework with classical general relativity theory.     

Objective properties from subjective quantum states: environment as a witness

We study the emergence of objective properties in open quantum systems. In our analysis, the environment is promoted from a passive role of a reservoir selectively destroying quantum coherence to an active role of amplifier selectively proliferating information about the system. We show that only preferred pointer states of the system can leave a redundant and therefore easily detectable imprint on the environment. Observers who-as is almost always the case-discover the state of the system indirectly (by probing a fraction of its environment) will find out only about the corresponding pointer observable. Many observers can act in this fashion independently and without perturbing the system. They will agree about its state. In this operational sense, preferred pointer states exist objectively.     

Quantum simulation of classical thermal states

We establish a connection between ground states of local quantum Hamiltonians and thermal states of classical spin systems. For any discrete classical statistical mechanical model in any spatial dimension, we find an associated quantum state such that the reduced density operator behaves as the thermal state of the classical system. We show that all these quantum states are unique ground states of a universal 5-body local quantum Hamiltonian acting on a (polynomially enlarged) qubit system on a 2D lattice. The only free parameters of the quantum Hamiltonian are coupling strengths of two-body interactions, which allow one to choose the type and dimension of the classical model as well as the interaction strength and temperature. This opens the possibility to study and simulate classical spin models in arbitrary dimension using a 2D quantum system.     

The quantization of the relativistic string

The classical field-dependent parametrization covariant Hamiltonian formulation of the open and the closed string is discussed. The formalism is not applicable to the open string. A conformally covariant formalism is developed for the open string. The Rohrlich gauge conditions are justified and applied. The parametrization of classical solutions is not uniquely fixed; the generators of rigid time translation in the parameter space remain first class. The constraints and gauge conditions are taken into account in the quantum theory as conditions on physical states. The required invariance of physical states under rigid displacement of parameter time leads to a mass superselection rule. The set of physical string quantum states is analogous to the set of states constructed by Di Vecchia, Del Guidice, and Fubini. A recursive construction is presented which permits the counting of physical states of any given mass, spin, and parity. Physical states lie on linearly rising Regge trajectories with one universal slope. The intercept of the leading trajectory is constrained only by the requirement that there be no tachyonic physical states. The quantization is carried out in four space-time dimensions.Supported by NSF Grant No. MPS74-15246 and DFG/Az 287/6. A portion of this work has been accepted by Syracuse University in partial fulfillment of the requirements of the doctorate degree.     

Classical breathers and quantum coherent states in discrete NLS systems

We present a comparison of quantum and “semiclassical” trajectories of coherent states that correspond to classical breather solutions of finite discrete nonlinear Schrödinger (DNLS) lattices. The main goal is to explain earlier numerical observations of recurrent return to the vicinity of initial coherent states corresponding to stable breathers that are also spatially localized. This effect can be considered as a quantum manifestation of classical spatial localization. We show that these phenomena are encoded in a simple expression for the distance between the quantum and semiclassical states that involves the basic frequencies of the classical and quantum systems, as well as the breather amplitude and quantum spectral decomposition of the system. A corollary is that recurrence phenomena are robust under perturbation of the initial conditions for stable breathers.     

Limit cycles in quantum theories

Renormalization group limit cycles and more chaotic behavior may be commonplace for quantum Hamiltonians requiring renormalization, in contrast to experience based on classical models with critical behavior, where fixed points are far more common. We discuss the simplest quantum model Hamiltonian identified so far that exhibits a renormalization group with both limit cycle and chaotic behavior. The model is a discrete Hermitian matrix with two coupling constants, both governed by a nonperturbative renormalization group equation that involves changes in only one of these couplings and is soluble analytically.     

The spectral gap for some spin chains with discrete symmetry breaking

We prove that for any finite set of generalized valence bond solid (GVBS) states of a quantum spin chain there exists a translation invariant finite-range Hamiltonian for which this set is the set of ground states. This result implies that there are GVBS models with arbitrary broken discrete symmetries that are described as combinations of lattice translations, lattice reflections, and local unitary or anti-unitary transformations. We also show that all GVBS models that satisfy some natural conditions have a spectral gap. The existence of a spectral gap is obtained by applying a simple and quite general strategy for proving lower bounds on the spectral gap of the generator of a classical or quantum spin dynamics. This general scheme is interesting in its own right and threfore, although the basic idea is not new, we present it in a system-independent setting. The results are illustrated with a number of examples.Copyright © 1994 by the author.FFaithful reproduction of this article by any means is permitted for non-commercial purposes.     

Evaluation of an algebraic model for the vibrations of water,effects of a discrete symmetry

We evaluate the dynamics of an algebraic model Hamiltonian for the vibrational motion of the water molecule. We pay special attention to the effects of the discrete symmetry of order 2 of the model. For a comparison between the quantum dynamics and the classical dynamics it is necessary to desymmetrize such quantum states which are based on types of motion which come in symmetry related pairs. For the other states based on motion invariant under the symmetry operation a desymmetrization would be meaningless. The desymmetrized quantum states show a simple connection to the guiding motions of the classical dynamics which can be used for a complete assignment of the states even though the system is not integrable in the sense of Liouville and shows chaotic behaviour in large parts of the classical phase space.     

Gauge covariant construction of the coherent states for a time-dependent harmonic oscillator by algebraic dynamical method

Time-dependent coherent states for a time-dependent harmonic oscillator are constructed in the framework of algebraic dynamics. These coherent states are gauge-covariant, and its time evolution is governed only by the solutions of a linear differential equation which describes the motion of the corresponding classical timedependent harmonic oscillator. Its non-classical and quantum statistical properties can thus be controlled by a proper choice of the frequency of the harmonic oscillator. Our coherent states reduce to Glauber coherent states in the case as the frequency is independent of time.     

Decoherence, time scales and pointer states

Certain issues regarding the time-scales over which environment-induced decoherence occurs, and the nature of emergent pointer states, are discussed. A model system, namely a Stern-Gerlach setup coupled to a quantum mechanical “heat-bath” is studied. The emergent pointer states for this system are obtained, which are different from those discussed in the literature. It is pointed out that this difference is due to some confusion regarding the decoherence time-scale, which is clarified here.     

Equilibrium states for a classical lattice gas

Various definitions of thermodynamic equilibrium states for a classical lattice gas are given and are proved to be equivalent. In all cases, a set of equations is given, the solutions of which are by definition equilibrium states. Examples are the condition of Lanford and Ruelle, and the KMS boundary condition. In connection with this, it is shown that the time translation for classical interactions exists as an automorphism of the quantum algebra of observables, under conditions which are weaker than those found for quantum interactions.     

Quantum Integrable Models and Discrete Classical Hirota Equations

The standard objects of quantum integrable systems are identified with elements of classical nonlinear integrable difference equations. The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. Elliptic solutions of Hirota's equation give a complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's Q-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to the Bethe ansatz are studied. The nested Bethe ansatz equations for A _k-1 -type models appear as discrete time equations of motions for zeros of classical τ-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation are analysed and a new determinant formula for eigenvalues of the quantum transfer matrices is obtained. Difference equations for eigenvalues of the Q-operators which generalize Baxter's three-term T−Q-relation are derived. Received: 15 May 1996 / Accepted: 25 November 1996